Annihilation creation operator

[greisen]

Hey,

I am calculating the expectation value of the position x. I have the wave function
psi(x) = |1> + |2>

so I use the equation <x> = <psi|(a+a^+|psi> to calculate the mean value. So I get

(<1| + <2|)(a+a^+)(|1> + |2>)

which I reduce to

<1|a|1> + <1|a|2> + <2|a|1> + <2|a|2> + <1|a+|1> + <1|a+|2> + <2|a+|1> + <2|a+|2>

if I use the properties of annihilation and creation I get some strange results such fx
sqrt(1) <1|0> or sqrt(3)<2|3> which are totally wrong. What have I done wrong?

Thanks in advance

 

[OlderDan]

Hey,

I am calculating the expectation value of the position x. I have the wave function
psi(x) = |1> + |2>

so I use the equation <x> = <psi|(a+a^+|psi> to calculate the mean value. So I get

(<1| + <2|)(a+a^+)(|1> + |2>)

which I reduce to

<1|a|1> + <1|a|2> + <2|a|1> + <2|a|2> + <1|a+|1> + <1|a+|2> + <2|a+|1> + <2|a+|2>

if I use the properties of annihilation and creation I get some strange results such fx
sqrt(1) <1|0> or sqrt(3)<2|3> which are totally wrong. What have I done wrong?

Thanks in advance

I am not following the notation. If a is an operator and a^ is an operator, then why is there a + after a^ and why are there no a^ in your expansion?

 

[greisen]

Maybe I have expressed the problem badly. a is the annihilation operator and a+ is the creation operator. I have a two state system |1>, |2> with a wavefunction (psi) = |1> + |2>. My problem is to perform the multiplication in order to find the expectation value:

(<1| + <2|)(a + a+)(|1> + |2>)

if I expand this calculation I get some strange results. How to multiple this? Any help appreciated - thanks in advance

 

[Euclid]

Hey,

<1|a|1> + <1|a|2> + <2|a|1> + <2|a|2> + <1|a+|1> + <1|a+|2> + <2|a+|1> + <2|a+|2>

if I use the properties of annihilation and creation I get some strange results such fx
sqrt(1) <1|0> or sqrt(3)<2|3> which are totally wrong. What have I done wrong?

Thanks in advance

<1|0>=<2|3>=0

In the above sum, only two terms are nonzero.

 

[OlderDan]

Maybe I have expressed the problem badly. a is the annihilation operator and a+ is the creation operator. I have a two state system |1>, |2> with a wavefunction (psi) = |1> + |2>. My problem is to perform the multiplication in order to find the expectation value:

(<1| + <2|)(a + a+)(|1> + |2>)

if I expand this calculation I get some strange results. How to multiple this? Any help appreciated - thanks in advance

So a|n> = |n-1> and a+|n> = |n+1>? and the states are orthogonal? Is that right? Euclid seems to know what you are doing, and it seems to be consistent with this interpretation. What are <1|1> and <2|2>? Are the states orthonormal? This is not a hint. I am asking because I want to know.

 

[greisen]

Yes the two kets |1> and |2> are orthonormale. My problem in the multiplication was the <2|3>, <1|0> which are zero.